We introduce a fiber bundle model where the interaction among fibers is modeled by an adjustable stress-transfer function that can interpolate between the two limiting cases of load redistribution, i.e., the global and the local load sharing schemes. By varying the range of interaction, several features of the model are numerically studied and a crossover from mean-field to short-range behavior is obtained. The properties of the two regimes and the emergence of the crossover in between are explored by numerically studying the dependence of the ultimate strength of the material on the system size, the distribution of avalanches of breakings, and of the cluster sizes of broken fibers. Finally, we analyze the moments of the cluster size distributions to accurately determine the value at which the crossover is observed.